You may be interested in attempts to salvage the Mary Rose, a Tudor-era warship that sank in the Solent in 1545. The standard way to raise sunken ships at the time (not very different from what we do today) was to pass cables under the hulk; attach those to two ships, pontoons, barges, or other lifting bodies; pull the cables taut with capstans at low tide; and wait for high tide to lift the ship. You can then tow the lifting bodies closer to shore until the hulk grounds, then repeat the process at succeeding low tides.

In 1545, though, the salvors ran into trouble with the Mary Rose wreck because the ship was lying on its side and had already become stuck in pretty deep silt. Without a modern diving apparatus, there wasn't any good way to pass cables under the ship, so the salvors attempted to right the ship by using the masts for leverage. This pretty much just resulted in the masts breaking off, and all they were able to recover were some guns, rigging and some other timber.

The Mary Rose Trust has a very nice Web page, if you want to learn more. The remains of the ship (which had deteriorated over time and eventually became covered in silt) were actually rediscovered in the 1970s, and the hull was raised in 1982.

The other ship from this era you're probably familiar with is the Swedish Vasa, which sank in a similar manner to how we think the Mary Rose sank, and was also attempted to be salvaged using the lifting body method. It was also salvaged in modern times, reaching the surface in 1961.

In addition to this, marine salvage using a diving bell or free-diving was a fairly common thing in the early modern period. A diving bell was used to salvage cannons from Vasa in 1658, for example (diving bells had been known of at least since Aristotle's time).