This is how We Dance; The Riverdance Fanlisting.

The Music of Riverdance

The music of Riverdance was composed by Bill Whelan. It was originally a 7-minute piece titled "Riverdance" and it was written for the 1994 Eurovision Song Contest where Riverdance was first performed by Michael Flatery and Jean Butler.
Later on, Bill Whelan wrote Riverdance The Show. After that, Riverdance: Music from the Show was released containing 15 original songs featuring Irish and International dancers like Eileen Ivers, Rafael Riquini, Nikola Parov and Davy Spillane. This record earned Bill Whelan a Grammy award in 1997.

To my knowledge, there are currently two Riverdance albums (three actually, but two of them are pretty much the same): Riverdance on Boradway, and Riverdance: Music from the Show. I've seen other albums around but they don't appear on Amazon or the Riverdance website.
Containing African-style songs, Riverdance on Broadway goes a bit further from the original Riverdance. Regardless, it is a great album and the lyrics to the songs found on that album can be seen below.

Lyrics

The Heart's Cry

Where the river foams and surges to the sea
Silver figures rise to find me
Wise and as daring
Following the heart's cry
I am that deep pool
I am that dark spring

Warm with a mystery
I may reveal to you
In time
(Time holds the heart's key)
Key to everything is Love
(Love makes the heart flower)
Flowers into a deep desire
(Passion in the heart's fire)
Passion and desire

See the eagle rise above the open plain
Golden in the morning air
Weaving and soaring
Watchful and protecting
I am your shelter
I will enfold you

Shivna

A Bhennáin, a bhúiredáin, a bhéicedáin bhinn
Is bhinn linn in cúicherán do ní tú 'sin ghlinn
A Bhennáin, a bhúiredáin, a bhéicedáin bhinn
Is bhinn linn in cúicherán do ní tú 'sin ghlinn

A dhraighnéin, a dhelgnacháin, (dil, dol, doh, roh day)
A mhinéin na chonaire (dil, dol, doh, ray)
A dhriséoc, a dhruimnechóc (dil, dol, doh, roh day)
A chaillech, a chuirrechennach in raga for ech?

A Bhennáin, a bhúiredáin, a bhéicedáin bhinn
Is bhinn linn in cúicherán do ní tú 'sin ghlinn

Is me Suibhne sirthechán; lúath reithim thar gleann
Nochan é m'ainm dligthechán; mó is ainm dam Fer Benn

At The Edge of the World

As we stand at the edge of the world
Our hearts will journey
For the Light at the edge of the world
Forever calls
And though our travels may be lost and wandering
From the dawn at the edge of the world
One voice calls to you

Fire may burn and the sky may thunder
Heroes crumble and the sun will fall
As the river circles on its endless journey
I will follow you.

As we stand at the edge of the world
There's a voice from the dawn of the world
As the rivers encircle the world
I will follow you.

Riverdance

Hear my cry
In my hungering search for you
Taste my breath on the wind
See the sky
As it mirrors my colours
Hints and whispers begin

I am living to nourish you, cherish you
I am pulsing the blood in your veins
Feel the magic and power of surrender to life
(Uisce Beatha)

Every finger is touching and searching
Until your secrets come out
In the dance, as it endlessly circles
I linger close to your mouth

Lift The Wings

How can the small flowers grow
If the wild winds blow
And the cold snow is all around

Where will the frail birds fly
If their homes on high
Have been torn down to the ground

Lift the Wings
That carry me away from here and
Fill the sail
That breaks the line to home
But when I'm miles and miles apart from you
I'm beside you when I think of you - a Stóirín a Grá

How can a tree stand tall
If a rain won't fall
To wash its branches down

How can the heart survive
Can it stay alive
If its love's denied for long
Lift the Wings
That carry me away from here and
Fill the sail
That breaks the line to home
But when I'm miles and miles apart from you
I'm beside you when I think of you - a Stóirín
And I'm with you when I dream of you - a Stóirín
And a song will bring you near to me - a Stóirín a Grá

Amanzi

BAWO! KUDALA SI GCEL 'INKULULEKO
(Lord! We ask for peace)
Si nje nge Limpopo
(We are like the Limpopo)

LIZO QALA NIN 'ITHEMBA LETHU?
(When will our hope begin)
Limpopo umfula wetu
(Limpopo our river)

BAWO! SIPHI ISITHEMBISO ?
(Lord! Where is the promise)
Re tshwana le Limpopo
(We are like the Limpopo)

ZIZO FIKA NINI ZONKE ?
(When will they all come)
Limpopo molapo wa rona
(Limpopo our beloved river)

AMANZI A YA PHEZULU EZIBHAKABHAKENI
(Water rises to the sky)
NJALO NATHI SI KHISHWA E MAKHAYENI
(In the same way we are taken from our homelands)

SI WELA NJENG 'IMVULA E MHLABENI
(Like rain, we fall on strange land)
SI HLAKAZWA E MOYENI
(We are scattered with the wind)

I Will Set You Free

In the dawn rain
In the noon fire
In the moon glow
Everywhere a story

In the parting glance
In the aching touch
In the tearful wave
Everywhere a tale

Give me one hand to reach for a vision
Give me one voice
I will make it sing
Give me one heart
I'll cradle it close
And with one dream
I will set you free
Yes with one dream
I will set you free.

On the dark roads
Of this strange land
Every eye stares
Every mouth is open

From the homeless days
To the heartless nights
When the hunger bites
Every door is closed

Give me one hand to reach for a vision
Give me one voice
I will make it sing
Give me one heart
I'll cradle it close
And with one dream
I will set you free

Give me one hand to reach for a vision
Give me one voice
I will make it sing
Give me one heart
I'll cradle it close
And with one dream
I will set you free
With my heart and soul
I will take you home.

Let Freedom Ring

Let freedom ring
Let freedom ring
From every mountaintop, those words of glory
From every heart and soul those voices sing
"Mama Africa, let Freedom ring"

Eye of wisdom
Hand of power
Gentle leader with a rebel heart
Lift your children
Heal their hunger
Bring back together what was torn apart

Let freedom ring
Let freedom ring
From every mountaintop, those words of glory
From every heart and soul those voices sing
"Mama Africa, let Freedom ring"

Throughout the homeland
There is a restlessness
There is a shiver and a wind of change
All around there are
Spirits rising
Reach for the moment. We can seize that day.

Let freedom ring
Let freedom ring
From every mountaintop, those words of glory
From every heart and soul those voices sing
"Mama Africa, let Freedom ring"

Endless Journey

Amazon the Liffey and the shimmer and the ripple
The Volga and the Fergus the Tagus and the Nile
Uisce in the Shannon and the Chico Colorado
Waikato Rio Grande the ripple in the tide

The gully and a gushing and an ebbing and a flowing
Irrawaddy Pilcomayo Mississippi and the Lee
Trickly-oozing in the Lagan Orinoco and Zambezi
Ubangi and the Congo the Mackenzie to the Sea

A pooling and a pouring and a flooding and a flushing
From the Purus to Parana from the Tigris to the Thames
Guadiana Guadalquiver Brahmaputra Colorado
And the Apalachicola-Chattahoochee and the Seine

Surging and a-showering and a-spouting and a-spilling
In the Albany the Oder the Ural and the Ooze
Dordogne and the Danube and Dnieper and the Dodder
Paraguay and Tokachi and Shinano Vera Cruz

Amazon the Liffey and the shimmer and the ripple
the Volga and the Fergus the Tagus and the Nile

The Heart will find a Home the Ship will find a Harbour
A haven for the soul until the storms subside
But when the warmer wind blows
And when that golden light glows
We face into the dawn and float out on the tide

Fire may burn and the sky may thunder
Heroes crumble and the sun may fall
As the river circles on its endless journey
I will follow you.

Amazon the Liffey and the shimmer and the ripple
the Volga and the Fergus the Tagus and the Nile
Uisce in the Shannon and the Chico Colorado
Waikato Rio Grande the ripple in the tide

Amazon the Liffey and the shimmer and the ripple
the Volga and the Fergus the Tagus and the Nile

We sail into the sun, our hope is the horizon
And just as we've begun, it seems that we've arrived
An everlasting journey -
A hungering and yearning
Around the river runs and rushes through our lives.

Fire may burn and the sky may thunder
Heroes crumble and the sun may fall
As the river circles on its endless journey
I will follow you.