I'm in Ireland, so we have a different system, but here my 3.5 year old hasn't been introduced to the concept of formal writing. In nursery, he learns the alphabet and numbers, and is shown what they look like, but nobody expects him to write.
At his developmental check a few weeks ago, he was asked to draw a circle. The nurse said all he was required to show was he could draw something attempting a closed loop.
He, however, does do a huge amount of play that is designed to help build pre-writing skills and hand-strength:
writing/messing in shaving foam
jelly play
threading beads
playing with tinned spaghetti (picking it up, moving it around)
linking coloured clothes pegs
lacing games
playdough - squashing and rolling it out
sandplay
Next year, when he's in pre-school he'll do much more about understanding letters and numbers (sandpaper cut outs are used), but he'll also do more things like spooning rice from one bowl to another, moving water from one container to another using a dropper, moving things using tweezers etc. Still, he won't write. His pencil grip will be worked out, and he'll draw lines to match number etc, but there's no expectation of him to write his name. If he does, great. If he doesn't no bother.
I don't get what the rush is. If a child can write their name at 3, then great, but if not, that's okay too. It's far better for a child to have the right foundations (hand strength, proper pencil grip, etc), than to just focus on the end result.