Can GFI breakers or GFI outlets allow grounded outlets to be used?
Yes, you can protect those outlets with a single GFCI per circuit and put three-prong outlets in the boxes. There will still be no ground wire and no actual grounding in those outlets.

(You cannot connect the ground terminal to the neutral; this is a code violation, and dangerous since it puts current on the ground wire, specifically causing the ground wire to act as a live conductor if there's a fault in the neutral, and will energize the screws on your face plates, energize the housings of grounded power tools and grounded appliances like toasters, etc.)

Technically, all un-grounded outlets that are protected with GFCI are supposed to be labeled "GFCI Protected/No Equipment Ground" (GFCI outlets will usually come with a little sheet of adhesive labels in the box for this purpose).

I agree generally with the comment about using GFCI breakers, although you certainly can replace the first outlet on a circuit with GFCI and protect everything "downstream" by connecting the downstream part of the circuit to the LOAD terminals on the GFCI.

Square D 20A GFCI breakers cost about $35 each. GFCI receptacles probably cost half that.

GFCI and grounding are two different concepts. GFCI's will protect people against electrocution, but they do not provide grounding, so they don't provide any protection to equipment that really requires grounding.

For some appliances this probably isn't an issue as the GFCI will provide protection against electrocution and there aren't other noticeable effects that matter to you.

For more sensitive electronics, like computers, TV's, radio and sound systems, running without a grounding conductor can subject your equipment to irritating (at best) effects like ground loop interference (which might sound like a background hum through speakers, for instance) and may or may not cause problems or damage to the equipment itself or even danger to your structure. If you have satellite equipment, there are explicit code requirements for grounding. Also, connecting surge protectors to a circuit without a ground probably won't allow the surge protector to be fully effective.

Inside the GFCI there are identical little inductive coils around the hot and the neutral conductors, and the GFCI circuitry trips if it detects any imbalance in the current between the hot and neutral. The neutral carries the same amount of current as the "hot" in a circuit with a load on it. If the neutral is not carrying the same amount of current, that means current is leaking to ground (a ground fault) through some other path, which might be a person. But the GFCI doesn't have anything to do with grounding the circuit.