Best Method for Injecting Around the Tear Trough Area?

Placement of filler in the periorbital is 3-dimensional art, with a little biology.
Intraoral or transcutaneous injections are used for placing hyaluronic acids in the perioral area. Because of the slightly greater precision and more favorable angle of inclination of the needle of transcutaneous injection, I usually prefer this for perioral injections, especially in patients with a prominent globe. One of the key vessels to the infraorbital periorbital fat is located right in the middle of the injection path from the intraoral approach.
Tyndall effects usually result from very superficial injection of hyualuronic acid, causing a discoloration. Hyaluronic acids should be injected in the epiperiosteal region.
Bruising can be minimized but not eliminated. There are some very large anatomically predictable vessels that occur in the periorbital area; in the performance of cheeklifts we see them daily and avoid them. Severe bruising can be related to damage to these vessels.

The tear trough area is extremely challenging as there are many individual anatomical factors at play, which influence the type of technique that an experienced practitioner will choose.
Bruising is a common mild side effect with tear trough fillers, and can be covered with camouflage make up.
Tyndall is the result of a large bolus of filler being placed superficially and bumps are due to large boluses of high G prime fillers placed deep. 
Treating the tear trough with fillers requires an individualised approach, light touch and an artists eye and "one size does not fit all"

Several points to consider with facial fillers in the periorbital zone.

Longevity is dependent on the type of filler and the quality of the recipient tissue.
Cross hatching minimizes bolus injections into the eyelids, thus minimizing the risk of granuloma formation or noticeable lumps.
Conservative injection may minimize overcorrection and allows the patient to acclimate to the fillers and the change in appearance.
Bruising is a natural consequence of injectables in zones where there are thin skin.
Bruising can be minimized by postprocedural ingestion of standardized Arnica such as Sinecch, BruiseKare. Or topical antibruising meds such as Bruisestick.
Experience is key in the area of the eyelids to avoid complications.
Results will manifest after several weeks and micromanagement of injection technique and postprocedural results are counterproductive.

There is no one technique that specifically is better than the other. It is the injector that makes the difference. It's like the bad golfer who can't hit the ball well with his clubs. He then gives them to a pro, and the pro kills the ball!  So, it is not the club it the clubber!

There is no outstanding technique that is absolutely the best. The most important factor is what is most comfortable and safe for the injector, physician or nurse, to consistently yield their best results. As you can tell from the other doctors’ answers, there are numerous technique available. And, there are specific times or situations where one certain technique may be more beneficial than another. In general, I tend to use more the percutaneous direct approach injecting just superficial to the tissue layer above the bone (periosteum) with conservative cross-hatching. Again, I think it is more important to look at focus more upon the results rather than the technique.

The injection of Restylane into the tear troughs is not a simple procedure and should only be done by a doctor experienced in surgery and anatomy of the eyelid. In my practice, I do all of the injections in the tear troughs and do not relegate these to my aesthetician. So, first of all, I would be sure you go to a plastic surgeon who is going to do the injection him or herself. This is more important that the "technique" used.
I prefer the transcutaneous route because of the angle of the needle and because I would not want to risk innoculating oral bacteria into the periorbital region and risk infection around the eye. This would be a disaster!
The procedure works best in the younger patients who have thicker, healthier skin in the lower lid than do older patients.
I prefer to inject just above the bone and put no traction on the tissues while I am injecting. As soon as the area I am in responds, I stop there and move further along the path of the tear trough. I take care not to insert the needle directly in the midpupillary line because of the blood vessels present there and their propensity for bruising. If I need to, I will do some massaging during the injections but I have learned to do as little as possible because it actually seems to create new areas that need injecting. Sometimes, a more superficial plane needs to be injected but this can lead to more bumpiness and bruising.
I have all patients avoid all aspirin, Motrin etc products for a week before the procedure and give them Arnica to take after. They are advised to avoid scheduling any significant social or work related events for a week because of the possibility of bruising.