To mount an Android phone via USB and get access to its files, you have to connect it, turn on the screen, swipe down your notifications, and hit a screen button. Install the doubleTwist Player (or another free app) to simply plug-and-transfer.
The previously mentioned doubleTwist player app for Android is intended to sync up nicely with the doubleTwist media manager on a PC or Mac, but you don't need to grab the "iTunes for Android" if you don't want. You don't even need to use the doubleTwist player on your Android—it's just the one free app we've found that automates the USB mounting process. Install it, and the next time you connect your Android phone to any computer, your microSD card will mount as any other USB storage would.
Update: A reader points us to a different app, suitably named "Auto Mount Your SD Card," that does the job for free and is in the Market (I swear, though, I searched using all of those terms at one point). As quite a few readers have their dislikes of the doubleTwist app's unannounced auto-mount, you can just install this app (QR code at left) and it only does that one thing.
Of course, not everybody may want this feature—for some, USB cords are more about charging than file access. You can unmount the card to resume simply charging, but if you'd like to disable the auto-mount feature, hit the Menu button inside the doubleTwist player app and un-check the feature, or simply un-install the app. Otherwise, you can search the Market to install the app, or use the QR code at the link.