According to The Encyclopedia of Chess Variants (ECV) by David B. Pritchard, chess variants based on this idea have been invented at least four times. The following descriptions are copied from ECV. (The names Battle Chess and Omni Chess have also been used for other chess variants. Swarm Chess was mentioned in a comment by Noam D. Elkies.)
BATTLE CHESS aka RATUSHNY'S GAME Origins unknown; quoted in a letter by Russell Chauvenet (Chess Jan 1944). Each player can, but is not obliged to, move every pawn and piece once in a turn. Multiple checks are possible. Tense play, with White holding a considerable advantage.
HURRICANE CHESS (1970s?) Every man may move once per turn. At no time during the turn may a player put himself in check. A player in check must get out of check on the first move of a turn.
OMNI CHESS Richard Fireman and Bob Gorga (1979). Each player has the right to move any number of his men once each on a turn. Castling counts as both K. and R. moves. Multiple checks are possible. A player need only escape check at the end of a turn. One of the first games played between the inventors:
White Richard Fireman Black Bob Gorga 1 (W) e4 d4 Bc4 Ne2 0-0 Qd3 a3 b4 f4 Bb2 Nc3 Rad1 (B) g6 Bg7 f5 e6 d6 Nf6 0-0 Nc6 a6 b5 Qe7 Bd7 Rae8 2 (W) exf5 Bxe6+ d5 g4 h3 Ne4 Nd4 Rde1 Rf3 Kh1 (B) gxf5 Bxe6 Nxd4 Nxd5 c6 Qh4 h6 Kf7 Rg8 3 (W) Nxd6+ Bxd4 c3 Rxe6 Qxf5+ Kg2 g5 Rd3 (B) Nxf4+ hxg5 Rxe6 Ke7 Bxd4 4 (W) Rxd4 Ne4 a4 Kf3 (B) bxa4 Qxh3+ g4+ Rh8 Rh6 Nd5 5 (W) Nc5 Rd3 Qe5+ Ke2 (B) Qg2+ Kf7 a3 Rh1 Rh3 6 (W) Rxh3 Kd3 Ne4 (B) gxh3 a2 Qf3+ Rd1+ Ne3 7 (W) c4 Qb2 Nd2 Kd4 (B) Rxd2+ h2 0-1
A rule that a player must escape check on the first move of a turn was later introduced. Despite the above example, it would seem that White has a substantial advantage. This could be offset by limiting White's moves on the first turn. This variant has been played by a number of masters.
SWARM CHESS Ralph Betza (1980) Every man that can move on a player's turn is obliged to do so. (There is no penalty if a man cannot move.) Moves can be played in any order. The opposing king can be captured as well as checkmated. A player, whose king may be subject to a number of checks, must get out of check on the first move of a turn. Castling is a king move (Nost-algia 248). The game overwhelmingly favours White. This could be compensated for by allowing White to move, say, only six men on the first turn.
[Nost-algia was the official organ of the defunct postal chess club NOST, the Knights of the Square Table.]