Ha! This is exactly why the race system is broken
Gringoes are just anglo-saxon peoples who are CALLED gringoes by the mexicans!
Elvish is easy, it sounds rather musical, and a bit breathy (like the wind is talking, no kidding..that is how I always describe elvish)
Dwarvish is VERY rough, tough, and grumpy. Lots of rolling r's just about everywhere.
Gringo would just speak english, lol, poor gringoes. You COULD have them speak french, or something akin to latin. That itsn't a bad idea.
Martian, like kito said, clicks, and pops, chirping and so on. Not a bad idea at all. You could come up with some interesting syntax for them.
Centuarian like I said earlier, something akin to german in sound and nature.
Space Wolf, to be honest they wouldn't truly have a whole lot to convey like we would because canines dont have the same makeup as we do when it comes to forming words or sounds. I could never figure out how the heck a Space Wolf could speak english...make a space communicator like in Start Trek? Anywho, lots of barking, nipping, yips, and howls. It's not too creative, but based upon their biology, ot's just about all I can think of from a professional point of view.
Fauns would probably poor something akin to greek since that is their origin anyhow.
Indians, as far as I know Sign Language was written by Spainards in the 17th Century actually. I don't know of Indians having an actual sign language system that provides them with a system to fully communicate. No, I would just give them something a little truer to their actual spoken language. I hate to say it, bt I'm never big on giving races more credit than their due, and Indians were certainly not a very mysterious people, nor very advanced.