I've got a pretty comprehensive answer here.
In addition to dealing with expressions like x => x.WeirdPropertyName, it can also deal with "extended" expressions such as x => x.WeirdMember.WeirdPropertyName.
Here's the code from that answer:
// code adjusted to prevent horizontal overflow
static string GetFullPropertyName<T, TProperty>
(Expression<Func<T, TProperty>> exp)
{
    MemberExpression memberExp;
    if (!TryFindMemberExpression(exp.Body, out memberExp))
        return string.Empty;
    var memberNames = new Stack<string>();
    do
    {
        memberNames.Push(memberExp.Member.Name);
    }
    while (TryFindMemberExpression(memberExp.Expression, out memberExp));
    return string.Join(".", memberNames.ToArray());
}
// code adjusted to prevent horizontal overflow
private static bool TryFindMemberExpression
(Expression exp, out MemberExpression memberExp)
{
    memberExp = exp as MemberExpression;
    if (memberExp != null)
    {
        // heyo! that was easy enough
        return true;
    }
    // if the compiler created an automatic conversion,
    // it'll look something like...
    // obj => Convert(obj.Property) [e.g., int -> object]
    // OR:
    // obj => ConvertChecked(obj.Property) [e.g., int -> long]
    // ...which are the cases checked in IsConversion
    if (IsConversion(exp) && exp is UnaryExpression)
    {
        memberExp = ((UnaryExpression)exp).Operand as MemberExpression;
        if (memberExp != null)
        {
            return true;
        }
    }
    return false;
}
private static bool IsConversion(Expression exp)
{
    return (
        exp.NodeType == ExpressionType.Convert ||
        exp.NodeType == ExpressionType.ConvertChecked
    );
}