Assuming there isn't a reason why it has to be the Google-copyrighted maps you use, I'd use OpenStreetMap for this (the open source wikipedia-style Google Maps alternative).

They're virtually identical but their license (Creative Commons Attribution-Sharealike) explicitly allows this sort of thing including for commercial work, so long as they're credited and you share the work you make with their data under the same licence, whereas Google are strict about usage of their maps unless you pay for an expensive license.

You can also output vectors, but the downside is, as is often the case with open source stuff, it's not slick and takes trial-and-error to get it all working...

Edit: the earlier version of this answer recommended using Open Street Map (OSM)'s export to PDF and SVG tools. I'd now recommend against fighting with these: they're junk. Not only are the files produced a mess (see edit history for examples - and the latest don't even open for me), but the tools almost never work, failing with a hopelessly misleading error message about "server load" being too high (<25), recommending trying again in a few minutes - but the server load is always far over 25 except for a few hours in European night time (and even then not all days).

There's a wiki with loads of alternate options, but all of these I've tried either don't work (many of them), don't work for anything except a close up of a few streets (MapOSMatic), give grainy pixel images in a PDF rather than actual vectors (Field Papers and Walking Papers), or where the "installation" process is more like configuring an entire server (most of them)...

There is however one process which seems to work:

1. Install Maperitive (free, simple install on Windows, requires a dependency on Mac / Linux).
2. Export the area of the map you want from Open Street Map. You might need to use one of the alternate links if it says the area is too large. Warning: for any area larger than a small town, files sizes are absurd. County level is 300mbs+. Countries will be many gigabytes. It gives you every single unnecessary detail at every zoom level below the area you specify, and there seems to be no facility to change this. Clearly no-one at OSM has ever heard of county maps, country maps or region maps.
3. Open it in Maperitive. You might need to change the file extension to .osm before it'll open
4. Change the rendering rules (what it shows and how) to ones which aren't insane.
   • You can get something pretty decent with Map > Switch to rules > googlemaps
   • Or, there's a huge amount of customisation you can do if you have the patience to tinker with the code with Map > Edit rendering rules
5. Use Map > Set printing bounds to set the area you want (it creates a near-invisible box filling the screen: zoom out and/or drag from a corner), and View > zoom to set the zoom level. Everything outside of the "printing bounds" will be cropped.
6. Export it with Tools > Export as SVG (Adobe Illustrator)

Note that if you want real vectors you must download then open the export from OSM rather than just browsing around in the default map that shows when you load up Maperitive - else it'll give you a fake SVG full of chopped up low quality bitmap tiles.

Another note - sometimes it doesn't actually update the file. If the file isn't changing, make sure you don't have it open in anything, and wait a few minutes between exporting and opening.