An "ODBC-Call Failed" error can be due to many things. The error message is usually accompanied by some further info which is more helpful, such as "Primary key violation." If you can include all the information it would help a lot. Nonetheless, here are my thoughts:
First, re-link the tables to make sure everything is current in Access
Errors while updating linked tables, in my experience, are usually due to invalid table joins, key (Primary or Foreign) violations, or data type mismatches. You don't mention whether your update queries involve multiple tables. If so, make sure the table joins are proper. If the joins are not on unique indexes, you'll have problems. The best way to ensure your update query is valid is to open a simple SELECT query (no updates) and view the records in datasheet view. Then, manually attempt to update one of the non-key values. If you can do that, you know your table joins and table links are OK. If not, then you've found the source of the problem.
Let us know if you can manually update the a row value from Access via your query -- that will get us to the next step.