Are you using Format> Borders & Shading? If so, that's the problem :-) The colors available there are not affected by Themes/Color Schemes -- the same 64 colors display in that matrix regardless of Theme or Color Scheme. Nor does that dialog update to reflect any custom choices you make by using the More Colors... button.
If you use the Shading button on the Tables tab of the Ribbon you'll see the the colors & shades available in the current Theme. That palette of colors will update when a different Theme or Color Scheme is chosen.
FWIW: I fully agree with John's position on the use of Table Styles, but they may not provide everything you want because they are tied to specific colors, not to Themes. IOW, if you select a different Theme/Color Scheme the user still has to edit the Table Style to match.
Further, If you're looking for the colors used in table to remain matched with colors used elsewhere in the document, that won't happen with Table Styles. For example, when shading changes in a Shape due to switching to a different Theme, shading in cells effected by a Style will not... it will remain whatever the style calls for. Likewise, applying a different Table style will not cause shading applied elsewhere to update.