If your new custom race is set to use voice that is named same as the new custom race :
Make a new quest in the quest tab area. Set its priority to 90. Leave the 'Start Game Enabled' box checked. Create a new condition under the main quest conditions, and set it to : GetIsRace MyNewCustomRace == 1 . Then all of your topics you put within that quest will only be shown and played for that race, you can also do this for each topic directly. In this entire case you do not need the race to be playable, it wont matter. Remember to drag in the GREETING topic for greetings.
If your new custom race is set to use a vanilla voice, like Imperial :
Do the same as above - but if you want your race to also play the pre-created Oblivion generic responses played (with full voice) for that race, then the race must be set as playable.
There are actually a few very good tutorials in the Cs Wiki regarding dialog, heres the best one I ever found, when I knew nothing about it all..
http://cs.elderscrolls.com/constwiki/index.php/Dialogue_Tutorial
(get to make a quest to get some mudcrab meat too!
)
As far as getting into actual new voice overs to work.. that is a whole new area, and again, can be found on the wiki.
Ps- I did a few things in my directions here that are just 'safe' to get things to work, like quest priority, Game Enabled, these settings are not mandatory, but are just safe for at least getting things to work off the get go.
Gl