New type of chess
There is something rather similar that already exists. Its a game called proteus by Steven Jackson games. Basically each side gets 8 pieces, but these pieces are dice that have represent a pyramid (stricty defensive piece, can't move or be taken) pawn, knight, bishop, rook, and queen. At the beggining of the game all your chessmen are pawns, with each turn you promote or demote one piece and move another. The game is over when a player can't move or has lost all his pieces. Then your pieces are scored based on value, Its really rather addictive. The rules can be seen here for free:
http://www.sjgames.com/proteus/

Tried playing any of the Total War games online? Plenty of wargames that do similar things, only difficulty would be finding opponents.
With the rules and 2 sets of ANY eight dice you could play test the game for yourself, but at only like $10 on amazon for the real game dice, it doesn't make a whole lot of sense to spend $4 on dice at wally world and paint half of them.
My wife and I play it quite a bit, we know the rules of real chess, never really studied it beyond that, but this game lets us play a faster game with just as much thought for our levels, and time with kids for leisure can be hard to come by.
I'd be very curious to hear opinions on the gameplay from people much stronger then myself.

I'd once envisioned a risk-like variation of chess where the success of a capture was determined by the roll of the dice. The same rules would apply, attackers getting the extra die but defenders winning the tie, and a peices point value (and perhaps mobility)would be used as a proxy for the "army count" in risk.
Once a peice had lost points during the battle, it would retain it's reduced point count and become more vulnerable as the game progressed.
It all seems kind of silly now that I've come to understand the incredible depth and balance in the game's current form though. I'm no longer big on chess variations as a result -- I think the game simply doesn't need changing.

There is something called "Gothic Chess"
http://en.wikipedia.org/wiki/Gothic_chess
It is played on a 10×8 board with two additional pawns per side and two fairy chess pieces:
a chancellor that moves as both a rook and a knight;
an archbishop that moves as both a bishop and a knight.
All standard chess rules are observed, including en passant capture and castling, but with the King moving three squares horizontally, when castling, instead of the usual two in standard chess. Gothic Chess has the following features:
- All of the pawns are defended in the starting position.
- There is a horizontal symmetry across the board regarding the pieces that have diagonal movements.
- The pieces on the four westernmost files and the three easternmost files are identical to chess, thus preserving many familiar patterns and motifs.
Id much rather play good ol' fashion chess though
I was thinking about various chess variations that could be constructed, yet still be fun, as well as intricate as original chess is now.
Borrowing the points system from some computer games, the variation could go like this:
As each piece is worth a certain amount of points, you are given 39 points to spend on your army. With this 39 points you can buy any units you choose and place them anywhere in the first two ranks. Obviously, both sides still have a king, and none of the pieces movement rules or worth is changed. So the queen = 9, rook = 5, knights and bishops = 3 and pawns 1. Also, you cannot have more units than there are spaces on the first two ranks, so no 39 pawn armies...
This would mimick real medieval battle strategy to a better extent. Forces were rarely composed of the same mix of troops.
So sometimes you could play the cavalry game, with 13 knights, like the Mongols who swept across Asia and Europe.
Or to mimick the English army at Agincourt - all archers... 13 bishops.
Or Elite regiment of 4 queens (and 3 leftover pawns).
Obviously, such unbalanced forces will not fare well and would need to be better balanced. So a front row of pawns would still be quite common.