Welcome back, thenostromo!
The ultimate origin of the financial senses of “check” seems to be that it originally referred to “the counterfoil of a bank bill, draft, etc.”, i.e., another piece of paper containing the same information, that could be used to check against forgery or alteration.
First OED2 cite in this sense is from 1706 (spelled “checque"). Bank drafts and payment orders typically had such a counterfoil and so by metonomy we get the term applied to the draft or payment order, i.e. the things in your checkbook, themselves. The leap from this to the bill in a restaurant, etc., is not covered in the OED2, but was discussed here.
Correction: they list the restaurant sense under a different headword, the sense of “something that stops, controls, or checks”. I think that Dave’s statement in the earlier thread, “The OED2 has all three coming from the sense of a counterfoil,” is a misreading of the entry. Rather, the counterfoil, the hat-check or coat-check token and the restaurant check are indicated to derive from (or be examples of) the sense “A means to ensure accuracy, correctness, security from fraud, etc.”