Virtualbox truns without VT enabled, it's software virtualization. It works faster on a VT enabled system, but it will also work without VT-x/AMD-V on the CPU.
All non-2011 "k" CPUs by Intel don't have virtualization, it's blocked by Intel, so don't even search for VT with a 3770k. Not all 2011-socket "k" CPUs have VT-x either, but some do.
The problem with Virtualbox for the moment is that their latest two versions are somewhat iffy. Not the VMWare is much better for the moment, but you might get lucky with VMPlayer instead of Virtualbox. If you're on linux, there are other options, but since you have a "k" CPU, I take it you're still on Windows.
Yes, it will accelerate software virtualization, but you can't use hardware virtualization (IOMMU). VT-x doesn't mean anything without VT-d if you're looking for a better virtualization alternative over Vitualbox/VMWare on Windows. You still have to get the software working. VT-x doesn't mean a whole lot without VT-d, because the essence of faster virtualization, whether software or hardware for that matter, is address translation between hardware. If VT-d is disabled, the graphics card/network card/etc... in the virtual environment won't be able to be accessed directly by the guest system, so you'll always have a slow running system, even if you don't use a full blown hardware virtualization system (kvm), and even if you want to use the next generation nVidia cards developed with the help of RedHat that will be able to compete with AMD's Mantle.
If you say so, why don't you explain it then mate ;-)
Anyway, there is a difference between hardware virtualization and software virtualization, you won't find a single 1155/1150-socket "k" CPU that has IOMMU, which is a prerequisite for kernel-based virtualization. Intel also uses different terms (VT, VT-x and VT-d) for the CPU extensions, you need a combination of 5 features to be able to virtualize efficiently: the virtualization extensions in the CPU (VT-x), the page table translation function (VT), the access to the on-die memory controller for IOMMU (VT-d), a chipset and PCI-e controller that allows for address translations, and a compatible kernel. With AMD, you need AMD-V (virtualization extensions and pagetable translation function), and AMD-Vi (IOMMU access to on-die memory controller), and there are no motherboards that I've heard of for AMD CPUs that have a PCIe controller that doesn't allow address translations, and the AMD chipsets don't handle address translations like Intel's, they are not a factor.