Driving a DC Motor without diodes
Well, what you're asking for has already been done for many years in a completely different field. We call "Diodeless H-bridges" class AB push pull amplifiers. You'll likely want some pretty fast transistors that have a low saturation voltage. Bonus, the less switching you do the more efficient this method is!

Schematic below:







simulate this circuit &ndash; Schematic created using CircuitLab