Have you heard about epigenetics? (epi, from the Greek, means above or over). It's the hottest thing, these days. What is it? It's modifications of genetic information, that can be heritable even without changing the sequence of the DNA. And this state is transmitted by non-Mendelian rules.
What the heck does that mean?
First of all, we've already talked about Mendelian inheritance, using eye color: each parent gives you one of their two alleles for eye color, so you get a complete set of two.
Turns out that our genes are packaged in the cell with proteins called histones. The histones can be chemically modified in ways that affects the expression of the genes that are associated with them. These modifications can occur at targeted regions of the genome (meaning, at the position of certain genes). Or they can occur randomly. They can be influenced by external factors (like the environment). or they can be hard-wired, genetically encoded (which seems circular but there we are...them's the facts!). And mechanisms exist to maintain these modifications when cells divide, or even from generation to generation. So the information carried by the modified histones can be inherited.
These epigenetic changes are ways to fine tune the genome. They don't change the sequence of the underlying DNA, but they may change how it's used. So even identical twins with exactly the same DNA can have different epigenetic signals. One study showed that genetically identical rats raised by a mother who groomed them grew up to express genes differently than their twins who were raised by a mother who ignored them. Seems the rats that were groomed were laid back, and epigenetic modifications switched off stress genes. The ungroomed rats were stressed out and neurotic, with those genes turned on.
In some cases (called imprinting) the same gene inherited from one parent is expressed, whereas when inherited from the other parent, it isn't. In the diagram, the green box means the gene is expressed, and the red box means it is shut off. I. and II. are siblings who express the allele they got from dad, and repress (turn off) the allele they got from mom. Note that it doesn't matter that I. and II. are different sexes. They express the paternal version of this gene, and silence the one from mom.
If this were to happen with eye color (it doesn't, but lets pretend for a minute it does), it would mean that you "shut off" whatever allele you got from Mom. So if mom gave you B, and dad gave you b, you'd still be blue-eyed, because you shut off the B from mom. Even though genetically you are Bb, you are not expressing the B gene, so you look as though you are bb.
But if Dad gave you a B, you'd express it, and be brown eyed. Identical genotype: Bb. Different outcome, depending on which parent gave you the gene.
Where it gets gnarly is what happens in the next generation. If you are a man (individual I. in the diagram), when you make sperm, they will code both the version you got from your dad AND the version you got from your mom as "paternal", and expressible. Because to your child, either one is paternal, they got it from you. As shown in the diagram section III., all the sperm are recoded as paternal. However, if you are a woman, your eggs will recode both as maternal, so they will be turned off in the offspring--section IV of the diagram, all the eggs are recoded as maternal, regardless of the source of the allele.
Using my eye color example, either the B or the b would be active in your child--so even if you were blue eyed yourself, you could legitimately have a brown eyed child.
These effects are laid over the top of the palette of genes that we inherit. You can't switch off a gene you never inherited. Even if you inherit a gene, it doesn't necessarily mean you express it. And, if you don't inherit the capability to switch off a gene, it doesn't matter how much mom licks your ears.
There a great Nova episode on epigenetics for more information.
To read this entire series in order, visit the Genetics Page.