Use the bus, or as the english call it coach. It used to be called Speedlink,takes 50-90 minutes and costs about $30. I have even experienced a beverage service on the bus!! you can board the bus and buy tickets in the ground transportation area.

I have a US size rollerboard and I'm always asked to gate check it. There is apparently a different European standard size for this which is of course much smaller. It seems that GA and FA see the extra size which immediatley draws attention to it.