As another bicyclist: yes. That's how we do it here in Tokyo. Almost no one rides on the roads, despite the painted bike lanes that are always ignored by cars and delivery trucks using them as parking spaces.
Another interesting thing about cycling in Tokyo that people from other countries may not be aware of is - bicycle insurance is mandatory. It costs like $2-$3/mo, and insures the medical costs of anyone you run into
As someone who just moved here: this is by far my least favorite part of moving around in the city; and having to dodge bikes on the sidewalks is the worst part of my commute.
Then stop dodging them: pedestrians like you are the worst. Just walk in a straight line where you're going and stop moving around so erratically; let the cyclists go around you.
Anyway, it would be nice if they built some nice, protected bike lanes, but they'd have to remove lanes from the cars to do that.
It reminds me of a mouse I once hit on my bike (not in Tokyo, it was a rural place). The mouse darted into the road, so I tried to go around him. But then he changed direction to avoid me, and so I ran over him. I ended up having to crush his skull to put him out of his misery. If he had just continued in a predictable, straight line, he would have survived.
You should rent one of the Docomo red rental bikes and ride it around some, and then you can see what it's like being one of the cyclists. The last thing you want as a cyclist is pedestrians who make unpredictable movements as you're passing. Also remember, pedestrians here have the absolute right-of-way, and cyclists are pretty much always at fault if there's a collision.