If you understand organic chemistry, biochemistry is (somewhat) conceptual and an application of your knowledge from orgo. If you don't understand organic chemistry, biochemistry is memorization. It's possible to get through biochem without a good understanding of organic, but having a good grasp of organic chemistry when going into biochem will give you a greater appreciation for the material.
One important set of topics from chemistry in general are the principles of chemical structure and bonding (bond geometry, stereochemistry, conformations, polarity, intermolecular forces). These concepts are crucial for understanding the structures of biological macromolecules (proteins, nucleic acids, lipids, and carbohydrates), which is central to undestanding the function of these molecules in cells. Knowing the properties and reactivities of functional groups (alcohols, carboxylic acids, amines, amides, imines, thiols, etc.) and understanding the basic reaction mechanisms in organic chemistry (what are nucleophiles? what are electrophiles?) will also be important for biochemistry. One big part of biochemistry is trying to understand how enzymes work, a subject which relies very much on arrow pushing from organic chemistry along with a discussion of protein structure. Knowing specific name reactions and their mechanisms from organic chemistry is not so important, but the ability to look at a reaction and think of a plausible reaction mechanism is important.
As agrobert said, unfortunately most introductory biochem is taught at a fairly low level and a good portion ends up being memorization (mostly the sections on metabolism). However, understanding the underlying chemistry that the enzymes in the metabolic pathways perform helps make learning these parts of metabolism easier.