I'm not certain what look you're after exactly without a pic as guidance but I'll suggest a couple of alternatives. Note: in both of the suggested approaches below the filled areas will become visible.
The first approach is to create a look similar to liming and you actually have a good start with the white paint you've already applied. You sand (or otherwise abrade, e.g. using Scotch-Brite or steel wool) until it exposes wood, with the white paint naturally left in the grain and recesses. Here are two examples of the effect:
If you're after a more uniform look than this will provide I'm afraid you're in for a lot more work. The first job would be to completely strip the furniture back to bare wood using chemical striper, heat gun + scraper or by sanding (I don't recommend the last method unless it's your only option).
After you're down to bare wood you would then brush on dilute paint, or apply the paint normally and then wipe away the excess with a cloth to leave a thin film. This can give slightly different effects depending on how thin the paint is, the colour of the wood etc. but here are two examples of how it can look:
With either method above it is advisable to apply a surface coating to help protect the furniture: the exposed wood in the first case and the thin, easily-damaged paint in the second case. Normal varnish is quite yellowish and will impart a slight amber tone to the white. If you don't mind the pieces becoming a sort of cream/ivory colour then go with a couple of thin wiped-on coats of regular oil-based polyurethane or other varnish.
To keep the colour as white as possible you need a 'water-white' finish, which in practice means either a waterbased varnish or lacquer. Waterbased poly is probably your simplest and cheapest option here.
Some people apply wax to the surface on this type of thing but it's really a very poor protective layer and I would only recommend it for a light-use item.