Actually, I think you'd find it easier to do in Illustrator than you think:
Create an "artboard" (what you and I would call a page in InDesign) the same size as the area you want to place in the background with InDesign. If you just want to covcer the page, do the same size. If you want to include bleed area, take that into account for your "artboard" before you start your work.
- Draw a vertical line along the left edge of the artboard. Make it the thickness and color you want it to be.
- Place a guide to the right by the distance you want for the grid, then Alt/Opt (depending on if you're working on a Windows PC or a Mac system, respectively)+Shift+drag a copy of the line to the guide you created. Alt/Opt makes the copy, Shift constrains the line to move perfectly horizontal to the first line. This creates a second line that lines up vertically by precisely the distance of your intended grid.
- Now comes the easy part: Press Ctrl/Cmd+D to repeat another line. The menu command would be Object>Transform>Transform Again. Lather-Rinse-Repeat with the Ctrl/Cmd+D thing until it steps and repeats perfectly across the artboard.
- Now draw a perfectly horizontal line across the top of the artboard. Adapt steps 2 and 3 as appropriate to apply the cross-hatching for your intended grid.
- Save the file. Place it in InDesign. Then marvel at your handiwork.
It's really pretty easy. And it should work for you just fine.
Good luck,
Randy