Kick and snare punch..

Compression is the key. The idea is to stretch out the drum sound by truncating the initial hit and then boosting the decay. One common way to do this is via parallel compression.
 
Any compressor can work for this, but some are geared specifically toward this application. One that will do the trick is a freebie from Nomad Factory called BusDriver. Or if you want to spend some money and get lots of options, check out Manic Compressor from Boz Digital Labs.
 
Make the attack short. Whether you want to completely suppress this initial stick hit or not depends on the effect you're after. I like to keep some stick, so set attacks around 2-3 ms. It helps to look at the waveform and calculate how long the initial transient actually is. Releases should also be short, enough that you clearly hear the tail being boosted. This may be in the 3 to 10 millisecond range, but it depends greatly on the instrument.
 
Manic Compressor also gives you an equalizer that can be applied pre or post, or to the sidechain. This can create a massively boomy kick drum if that's what you want.
 
If you really want to get fancy, a multi-band compressor can be used to accentuate different elements within the drums' sounds. If you want a clicky kick for example, you can make the compressor only affect the drum's low frequencies. Or beef up the snares on a snare drum, or get a Ringo-style ride wash.
 
Finally, gating can help drum thickness. Since you mention Def Leppard, chances are some of that is the classic gated reverb effect. Whether you use reverb or compression to bring up drum tails and room ambiance, a negative side-effect is that hits can become indistinct and run into one another. Gating cuts off the drum after a period calculated from the project tempo. If you exaggerate this effect, you get a classic 80's snare sound; but do it subtly and get the timing right, the effect can be quite transparent.