Somebody posted a portable client, but I don't trust them, so I'll show you how to make your own.
Windows guide
Download the windows .zip from bitcoin.org
Unpack it. Go into that directory.
Make a file called launch.txt
Put this in it:
start bitcoin-qt.exe -datadir=data
Rename it to launch.bat
Make a folder called data
It should look like this. Double-click launch.bat
to start it. If files show up in the data folder you made, then you've done things correctly.
Macintosh guide
Download the Mac .dmg from bitcoin.org
Find the Bitcoin-Qt.app file, and copy it to to your flash drive.
Go to Launchpad, type in 'terminal'. Hit enter.
When the terminal comes up, type:
cd /path/to/flash/drive
mkdir data
cd Bitcoin-Qt.app/Content/MacOS
nano start.sh; chmod +x start.sh
Type in:
#!/bin/bash
dir1=$(cd "$(dirname "$0")"; pwd)
dir=`echo $dir1 | sed 's/"/\\"/g' | tr -d '\n'`
data="$dir/../../data"
"$dir"/bitcoin -datadir="$data"
Now hit Ctrl-O, enter, Ctrl-X. You should be back at the terminal.
cd ..
nano Info.plist
Find the lines that look like this:
<key>CFBundleExecutable</key>
<string>Bitcoin-Qt</string>
Change it to:
<key>CFBundleExecutable</key>
<string>start.sh</string>
Now hit Ctrl-O, enter, Ctrl-X. You should be back at the terminal. Now we just need to give it a better name.
cd ../..
mv Bitcoin-Qt.app BitcoinPortable.app
Try running your BitcoinPortable.app. If you see files show up in the data directory next to it, then you've done things correctly!
Thanks to Pygy for the script
Note: I don't have a Mac, so this is untested.