Votive relief from the temple of Artemis at Brauron
There is a subtile difference between votives and thank-offerings: votives were gifts promised to a Theos or Theia in advance--so in relation to a vow made to a Theos or Theia--while thank-offerings were given in relation to an unexpected event, or as a sign of long-term devotion. Two examples:
A Hellenic woman will give birth in the following days. This is a dangerous affair, and so she visits the temple of Artemis to beg for a swift and safe delivery. She vows that, when the child is born and both she and the child are healthy, she will have a relief crafted to honor Artemis. When both mother and child survive the birth, the miasma cause by said birth have worn off, and the relief is completed, the mother brings the relief to the temple and presents it to the priestess, who accepts on behalf of the Goddess. The mother's debt is now repaid. This is a votive offering.
In another example, a blacksmith is finally ready to retire. His son has taken over the family business and the man takes his beloved tools to the temple of Hēphaistos, where he leaves them as an offering of gratitude for the long years of inspiration, aid and skill the God has provided him. This is a thank-offering.
A special type of votives were pinakes (πίνακες, singular: πίναξ). A pinax was a votive tablet of painted wood, terracotta, marble or bronze that was deposited in a sanctuary or placed within a burial chamber as a memorial. Most pinakes were gifted to the temple as votive offerings, and depicted scenes of libations, mythological scenes, scenes from daily life (like farming or household duties). Another type of scene was the depiction of body parts. These pinakes were gifted as a call for healing aid, and were most likely gifted before the actual healing took place, something unique to pinakes. Lokris (Λοκρίς), a region in ancient Hellas, is known for a remarkable archeological find of thousands of pinakes, most of them from sanctuaries of Persephone and Aphrodite.
Both votives and thank-offerings were called dedications, and could be anything. Money was rarely given; if a merchant promised ten percent of his profit to Hermes for keeping him safe on the road, the merchant would ask a local craftsman to create a statue, relief, plaque, plate or other item which he would be able to donate to the temple. The temple would then display the item either inside of the temple, in a separate room, called a naos (ναός), on the property, called the temenos (τέμενος), or in a specially constructed treasury: a large and finely crafted building which was solely used to house treasures.
Important to note is that the entire temenos--often identified with a low wall or fence--was holy. Anything on that property was sacred to the Theos or Theia in question. To steal anything from the temenos was a civil crime, as well as a crime to the Theos or Theia--who would punish the thief severely. Fear for this punishment and respect for the Theoi was so high that temples, treasuries, and entrance gates to their temenos were hardly ever--if ever--locked.
Once dedications were offered to the Theos or Theia and the temple, they remained sacred, even if they broke, spoiled, wasted away or otherwise became impossible to display. When this happened the priest or priestess of the temple would take the remains of the dedication and bury it in a votives-pit, located somewhere on the temenos.
So, what does this mean for Hellenics who do not have a temple at their disposal? From the ancient dedications--many of which can be found in museums today--we can take three lessons: statuary wasn't used to represent a Theos or Theia, but to honor or thank Him or Her; dedications, once broken or used up, were still sacred to the Theoi, and were disposed of in a respectful manner, on the sacred site; and dedications were an important part of Hellenic practice, and thus they should be so in Hellenismos.
It may be prudent to assign a specific part of your garden as a disposal site; a place surrounded by sacred plants and herbs, well kept, but never turned over for planting. A place away from where children make mud pies.
As for dedications: buying or making statuary, miniatures of sacrificial or sacred animals, or other non-perishable offerings for the Theoi you worship and presenting them on the sacred or festival days of the Theos or Theia in question would be good practice. Presenting Them with something like that as a vow or thank-offering also seems perfectly acceptable. For those not in a position to give these types of dedications, one can always offer incense, wine, or other perishables associated with the Theos or Theia one wants to thank or vow to. Remember: talk is cheap; Hellenismos is a religion of action. Not following through on a vow made to a Theos or Theia is dangerous practice, indeed.