Is it safe to substitute a 10-gauge wire for ground strap on dryer?
All due respect to Tester101, I'm not a fan of this method for a couple of reasons.  

Not a listed method

And that matters because a failure here is lethal.  A lost neutral is precisely the failure that spooks us and is why NEMA 10 is considered bad news. 

A break in a neutral wire will cause (internal to the appliance) the neutral to be "pulled up" to one of the hots.  That is normal in any appliance.  The problem is, by booting ground to neutral, the chassis of the machine is now hot. That problem is doubled because the washing machine right next to it is still properly grounded.  Touch both, you die. 

This is literally bootlegging ground.  It is illegal in almost every context, (except this context, only due to lobbying by appliance manufacturers.)

You might as well do the best thing

Instead of running the grounding wire 2 inches to the neutral lug, you could run it 20 feet... to a proper grounding point. 

OK, The right right way is to retrofit a ground: run it to the receptacle, change the receptacle for NEMA 14 and use that to support a NEMA 14 receptacle and plug.