I use Mailwasher to check my webmail accounts every minute. I know that if the site goes down then I get an error in the mail checking.

... but I also know that sometimes webmail can go down or have a problem, but the site is up and running fine.

Is there something similar that I can use to check my sites every minute (or other period) and let's me know within that period if it is down?

Here are some free resources. There are also a number of paid services and applications available.

Basicstate.com is an free online monitor.
free web site uptime monitor service (http://basicstate.com/)

Pingability is another free, but highly limited service.
Pingability.com: Web Site Monitoring and Alert Service (http://pingability.com/)

Pingdom is a distributed computing based application.
Pingdom GIGRIB - Free Software Downloads and Software Reviews - Download.com (http://www.download.com/Pingdom-GIGRIB/3000-2017_4-10648938.html?tag=lst-0-3)

Expert Website Monitor provides a lot more information other than uptime status.
Expert Website Monitor - Free Software Downloads and Software Reviews - Download.com (http://www.download.com/Expert-Website-Monitor/3000-2085_4-10629850.html?tag=lst-0-10)

Expert Webmaster is similar to Expert Website Monitor, but provided even more information.
Expert Webmaster - Free Software Downloads and Software Reviews - Download.com (http://www.download.com/Expert-Webmaster/3000-2085_4-10563850.html?tag=lst-0-4)